Q1E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

In Problems 1-10, use a substitution y=x^r to find the general solution to the given equation for x>0.

x²y"(x)+6xy'(x)+6y(x)=0

Step-by-Step Solution

Answer

The general solution to the given equation x²y"(x)+6xy'(x)+6y(x)=0 is y=c₁x^-2+c₂x^-3.

Define Cauchy-Euler equations:

In mathematics, a Cauchy problem is one in which the solution to a partial differential equation must satisfy specific constraints specified on a hypersurface in the domain. An initial value problem or a boundary value problem is both examples of Cauchy problems. The equation will be in the form of ax²y"+bxy'+cy=0.

Find the general solution:

Given, 

x²y"(x)+6xy'(x)+6y(x)=0 

Let L be the differential operator defined by the left hand side of the equation.

L[y](x)=x²y"+6xy'+6y

w(r,x)=x^r

By substituting you get,

L[y](x)=x²(x^r)"+6x(x^r)'+6x^r 

=x²(r(r-1)) x^r-2+6x(r)x^r-1+6x^r  

=(r²-r) x^r +6rx^r +6x^r

=(r²+5r+6) x^r

Solving the indicial equation.

r²+5r+6 =0

(r+3) (r+2)=0

The two distinct roots are,

r₁=-2

r₂=-3

There are two linearly independent solutions.

y₁=c₁x^-2

y₂=c₂x^-3

The general solution is y=c₁x^-2+c₂x^-3.